Partial permutation decoding for binary linear Hadamard codes

Permutation decoding is a technique which involves finding a subset S, called PD-set, of the permutation automorphism group PAut(C) of a code C in order to assist in decoding. A method to obtain s-PD-sets of size s + 1 for partial permutation decoding for the binary linear Hadamard codes H m of length 2 m , for all m ≥ 4 and 1 < s ≤ (2 m − m − 1)/(1 + m) , is described. Moreover, a recursive construction to obtain s-PD-sets of size s + 1 for H m+1 of length 2 m+1 , from a given s-PD-set of the same size for the Hadamard code of half length H m is also established

PD-sets for (nonlinear) Hadamard Z₄-linear codes

Publicació amb motiu de la 21st Conference on Applications of Computer Algebra (July 20-24, 2015, Kalamata, Greece

PD-sets for Z₄-linear codes : Hadamard and Kerdock codes

Permutation decoding is a technique that strongly depends on the existence of a special subset, called PD-set, of the permutation automorphism group of a code. In this paper, a general criterion to obtain s-PD-sets of size s+1, which enable correction up to s errors, for Z₄-linear codes is provided. Furthermore, some explicit constructions of s-PD-sets of size s+1 for important families of (nonlinear) Z₄-linear codes such as Hadamard and Kerdock codes are given

Partial permutation decoding for binary linear and Z4-linear Hadamard codes

In this paper, s-PD-sets of minimum size s + 1 for partial permutation decoding for the binary linear Hadamard code H_m of length 2^m , for all m ≥ 4 and 2 ≤ s ≤ floor(2^m/(1+m)) -1, are constructed. Moreover, recursive constructions to obtain s-PD-sets of size l ≥ s + 1 for H_{m+1} of length 2^(m+1), from an s-PD-set of the same size for H_m , are also described. These results are generalized to find s-PD-sets for the Z4 -linear Hadamard codes H_{γ,δ} of length 2^m , m = γ + 2δ − 1, which are binary Hadamard codes (not necessarily linear) obtained as the Gray map image of quaternary linear codes of type 2^γ 4^δ . Specifically, s-PD-sets of minimum size s + 1 for H_{γ,δ} , for all δ ≥ 3 and 2 ≤ s ≤ floor(2^(2δ−2)/δ)-1, are constructed and recursive constructions are described

Partial permutation decoding for binary linear Hadamard codes

Permutation decoding is a technique which involves finding a subset S, called PD-set, of the permutation automorphism group PAut(C) of a code C in order to assist in decoding. A method to obtain s-PD-sets of size s + 1 for partial permutation decoding for the binary linear Hadamard codes H m of length 2 m , for all m ≥ 4 and 1 < s ≤ (2 m − m − 1)/(1 + m) , is described. Moreover, a recursive construction to obtain s-PD-sets of size s + 1 for H m+1 of length 2 m+1 , from a given s-PD-set of the same size for the Hadamard code of half length H m is also established